In this paper, we shall prove a Carleman estimate for the so-called Zaremba problem. Using some techniques of interpolation and spectral estimates, we deduce a result of stabilization for the wave equation by means of a linear Neumann feedback on the boundary. This extends previous results from the literature: indeed, our logarithmic decay result is obtained while the part where the feedback is applied contacts the boundary zone driven by an homogeneous Dirichlet condition. We also derive a controllability result for the heat equation with the Zaremba boundary condition.